By conventional methods we need to know or look up the appropriate formula:

We substitute the four values, simplify, remove the fraction, open the brackets and rearrange the equation to finally get 3y = -4x + 29.

Or, by the one-line Vedic method:

By vertically and crosswise:

we subtract vertically in the first column to get the y-coefficient, 5 – 2 = 3,

we subtract vertically in the second column to get the x-coefficient, 3 – 7 = -4,

and we cross-multiply and subtract to get the absolute term, 5×7 – 3×2 = 29.

We can also solve all sorts of problems in coordinate geometry, transformations, trigonometry etc. and there are more advanced applications in 3-dimensional work, trigonometrical equations, differential equations, complex numbers, simple harmonic motion and so on.

In addition to the general methods described above the Vedic system offers many special methods which can be used when certain conditions are satisfied. These are often extremely effective and powerful. The final example is a special method.